Credit Swaps
Credit Default Swaps
1
Generic Credit Default Swap: Definition

In a standard credit default swap (CDS), a
counterparty buys protection against default by a
particular company or economic entity from a
counterparty (seller).

The company or entity is known as the reference
entity and a default by that entity is known as a
credit event.

The buyer of the CDS makes periodic payments or a
premium to the seller until the end of the life of the
CDS or until the credit event occurs.
2
Generic Credit Default Swap: Definition

Depending on the contract, if the credit event
occurs, the buyer has
 The right to sell a particular bond (or loan) issued
by the company for its par value (physical delivery)
Or
 Receive a cash settlement based on the difference
between the defaulted bond’s par value and its
market price (recovery value)
3
Generic Credit Default Swap: Definition

Example:
Suppose two parties enter into a 5-year CDS with a
NP of $200,000,000.

The buyer agrees to pay 95 bp annually for
protection against default by the reference entity.

If the reference entity does not default, the buyer
does not receive a payoff and ends up paying
$1,900,000 each year for 5 years.

If a credit event does occur, the buyer will receive
the default payment and pay a final accrual payment
on the unpaid premium.
4
Generic Credit Default Swap: Definition
Example:

Note:

If the event occurs half way through the year, then the buyer pays
the seller $950,000.

If the swap contract calls for physical delivery, the buyer will sell
$200 million par value of the defaulted bonds for $200,000,000.

If there is a cash settlement, then an agent will poll dealers to
determine a mid-market value. If the recovery value were $30 per
$100 face value, then the buyer would receive $140,000,000 minus
the $950,000 accrued interest payment.
5
CDS Terms
1.
In the standard CDS, payments are usually made in
arrears either on a quarter, semiannual, or annual basis.
2.
The par value of the bond or debt is the notional
principal used for determining the payments of the
buyer.
3.
In many CDS contracts, a number of bonds or credits
can be delivered in the case of a default.

A company like Motorola, for example, might have 10 bonds
with similar maturities, coupons, and embedded option and
protection features that a buyer of a CDS could select in the
event of a default.
6
CDS Terms
4.
In the event of a default, the payoff from the CDS is equal to the
face value of the bond (or NP) minus the value of the bond just
after the default.

The value of the bond just after the default expressed as a
proportion of the bond’s face value is known as the recovery
rate (RR).
CDS Payoff = (1 − RR)NP – Accrued Payment

If that value on the $200,000,000 CDS were $30 per $100 face
value, then the recovery rate would be 30% and the payoff to
the CDS buyer would be $140,000,000 minus any accrued
payment.
Payoff = (1 −.30)$200,000,000) − Accrued Payment
Payoff = $140,000,000 − Accrued Payment
7
CDS Terms
5.
The payments on a CDS are quoted as an annual
percentage of the NP.

The payment is referred to as the CDS spread.
8
CDS Terms
6.
Swap bankers function as both brokers and dealers
in the CDS market.
 As dealers, they will provide bid and ask quotes
on a particular credit entry.
 For example, a swap bank might quote a 5-year
CDS on a GE credit at 270 bp bid and 280 bp
offer.

The swap bank will buy protection on GE for 2.7% of
the underlying credit’s principal per year for 5 years

The swap bank will sell protection on GE for 2.8% of
the principal.
9
CDS Uses
 CDS are used primarily to manage the
credit risk on debt and fixed-income
investment positions.
10
CDS Uses
Example 1:

Consider a bond fund manager who just purchased a
5-year BBB corporate bond at a price yielding 8%
and wanted to eliminate the credit risk on the bond.

To eliminate default risk, suppose the manager
bought a 5-year CDS on the bond.

If the spread on the CDS were equal to 2% of the
bond’s principal, then the purchase of the CDS
would have the effect of making the 8% BBB bond
a risk-free bond yielding approximately 6%.
11
CDS Uses
Example 1:

That is, if the bond does not default, then the bond
fund manager will receive 6% from owning the bond
and the CDS (8% yield on bond – 2% payment on
CDS).

If the bond defaults, then the bond manager would
receive 6% from the bond and CDS up to the time of
the default and then would receive the face value on
the bond from the CDS seller, which the manager can
reinvest for the remainder of the 5-year period.

Thus, the CDS allows the manager to reduce or
eliminate the credit risk on the bond.
12
CDS Uses
Example 2:

Suppose a manager holding a portfolio of 5-year
U.S. Treasury notes yielding 6% expected the
economy to improve and therefore was willing to
assume more credit risk in return for a higher return
by buying BBB corporate bonds yielding 8%.

As an alternative to selling his Treasuries and
buying the corporate bonds, the manager could sell a
CDS.
13
CDS Uses
Example 2:

If he were to sell a 5-year CDS on the above 5-year
BBB bond to a swap bank for the 2% spread, then the
manager would be adding 2% to the 6% yield on his
Treasuries to obtain an effective yield of 8%.

Thus with the CDS, the manager would be able to
obtain an expected yield equivalent to the BBB bond
yield and would also be assuming the same credit risk
associated with that bond.
14
CDS Uses

Example 3:
Consider a commercial bank with a large loan to a
corporation.

Prior to the introduction of CDS, the bank would
typically have to hold on to the loan once it was
created.

During this period, its only strategy for minimizing its
loan portfolio’s exposure to credit risk was to create a
diversified loan portfolio.
15
CDS Uses

Example 3:
With CDS, such a bank can now buy credit
protection for the loan.

In general, CDSs allow banks and other financial
institutions to more actively manage the credit risk
on their loan portfolio, buying CDSs on some loans
and selling CDSs on other.

Today, commercial banks are largest purchasers of
CDS and insurance companies are the largest sellers.
16
The Equilibrium CDS Spread
17
The Equilibrium CDS Spread
 In equilibrium, the payment or spread on a
CDS should be approximately equal to the
credit spread on the CDS’s underlying bond
or credit.
18
The Equilibrium CDS Spread
Example:

If the only risk on a 5-year BBB corporate bond
yielding 8% were credit risk and the risk-free rate on
5-year investment were 6%, then the bond would be
trading in the market with a 2% credit spread.
19
The Equilibrium CDS Spread
Example:

If the spread on 5-year CDS on a BBB quality bond
were 2%, then an investor could obtain a 5-year
risk-free investment yielding 6% by either
Buying a 5-year Treasury
or
Buying the 5-year BBB corporate yielding 8%
and purchasing the CDS on the underlying
credit at a 2% spread
20
The Equilibrium CDS Spread

Example:
If the spread on a CDS is not equal to the credit
spread on the underlying bond, then an arbitrage
opportunity would exist by taking positions in the
bond, risk-free security, and the CDS.
21
The Equilibrium CDS Spread

CDS Spread = 1% < Credit Spread = 2%
Suppose a swap bank were offering the above CDS for
1% instead of 2%.

In this case, an investor looking for a 5-year risk-free
investment would find it advantageous to create the
synthetic risk-free investment with the BBB bond and
the CDS.

That is, the investor could earn 1% more than the yield
on the Treasury by creating a synthetic treasury by
1. Buying the 5-year BBB corporate yielding 8%
and
2. Purchasing the CDS on the underlying credit at a 1%
22
The Equilibrium CDS Spread
CDS Spread = 1% < Credit Spread = 2%

If the swap bank were offering the above CDS for
1% instead of 2%, then an arbitrager could realized
a free lunch equivalent to a 5-year cash flow of 1%
of the par value of bond by
1. Shorting the Treasury at 6% (or borrowing at
6%)
2. Using the proceeds to buy the BBB corporate
3. Buying the CDS
23
The Equilibrium CDS Spread
CDS Spread = 1% < Credit Spread = 2%
 These actions by investors and arbitrageurs,
in turn, would have the impact of pushing
the spread on the CDS towards 2%—the
underlying bond’s credit spread.
24
The Equilibrium CDS Spread

CDS Spread = 3% > Credit Spread = 2%
If the swap bank were offering the CDS at a 3%
spread, then an investor looking for a 5-year riskfree investment would obviously prefer a 6%
Treasury yielding 6% to a synthetic risk-free
investment formed with the 5-year BBB corporate
yielding 8% and a CDS on the credit requiring a
payment of 3%.
25
The Equilibrium CDS Spread

CDS Spread = 3% > Credit Spread = 2%
If the swap bank were offering the CDS at a 3%
spread, then a more aggressive investor looking to
invest in the higher yielding 5-year BBB bonds,
though, could earn 1% more than the 8% on the BBB
bond by creating a synthetic 5-year BBB bond by
1. Purchasing the 5-year Treasury at 6%
2. Selling the CDS at 3%
26
The Equilibrium CDS Spread

CDS Spread = 3% > Credit Spread = 2%
If the swap bank were offering the CDS at a 3%
spread, then a bond portfolio manager holding 5-year
BBB bonds yielding 8% could pick up an additional
1% yield with the same credit risk exposure by
1. Selling the BBB bonds
2. Selling the CDS at 3%
3. Using the proceeds from the bond sale to buy the
5-year Treasuries yielding 6%
27
The Equilibrium CDS Spread
CDS Spread = 3% > Credit Spread = 2%

If the swap bank were offering the CDS at a 3%
spread, then an arbitrager could realized a free lunch
equivalent to a 5-year cash flow of 1% of the par
value on the bond by
1. Shorting the BBB bond
2. Selling the CDS
3. Using proceeds from bond sale to purchase 5year Treasuries
28
The Equilibrium CDS Spread

CDS Spread = 3% > Credit Spread = 2%
With these positions, the arbitrageur would receive for
each of the next 5 years 6% from her Treasury investment
and 3% from her CDS, but only pay 8% on her short BBB
bond position.

Furthermore, her holdings of Treasury securities would
enable her to cover her obligation on the CDS if there was
a default.

That is, in the event of a default she would be able to pay
the CDS holder from the net proceed from selling her
Treasuries and closing her short BBB bond by buying
back the corporate bonds at their defaulted recovery price.
29
The Equilibrium CDS Spread

CDS Spread = 3% > Credit Spread = 2%
Collectively, the actions of the investors, bond
portfolio managers, and arbitrageur would have the
effect of pushing the spread on CDS from 3% to
2%.
30
The Equilibrium CDS Spread

In equilibrium, arbitrageurs and investors should
ensure that the spreads on CDS are approximately
equal to spreads on the underlying bond or credit.

This spread can be defined as the equilibrium
spread and is referred to as the arbitrage-free
spread and the Z-spread.
31
CDS Spread and the Expected Default Loss
 The arbitrage-free spread, Z, on a bond or
CDS can also be thought of as the bond
investor’s or CDS buyer’s expected loss
from the principal from default.
32
CDS Spread and the Expected Default Loss

Consider a portfolio of 5-year BBB bonds trading
at a 2% credit spread.

The 2% premium that investors receive from the
bond portfolio represents their compensation for an
implied expected loss of 2% per year of the
principal from the defaulted bonds.
33
CDS Spread and the Expected Default Loss

If the spread were 2% and bond investors believed
that the expected loss from default on such bonds
would be only 1% per year of the principal, then the
bond investors would want more BBB bonds, driving
the price up and the yield down until the premium
reflected a 1% spread.

If the spread were 2% and bond investors believed
the default loss on a portfolio of BBB bond would be
3% per year, then the demand and price for such
bonds would decrease, increasing the yield to reflect
a credit spread of 3%.
34
CDS Spread and the Expected Default Loss

Thus, in an efficient market, the credit spread on
bonds and the equilibrium spreads on CDS represent
the market’s implied expectation of the expected loss
per year from the principal from default.

In the case of a CDS, the equilibrium spread can
therefore be defined as the implied probability of
default of principal on the contract.
35
CDS Valuation
36
CDS Valuation
 The total value of a CDS’s payments is equal to
the sum of the present values of the periodic CDS
spread (Z) times the NP over the life of the CDS,
discounted at the risk-free rate (R):
M
Z NP
PV(CDS Payments)  
t
(
1

R
)
t 1
37
CDS Valuation
 In terms of the above example, the present value of
the payment on the 5-year CDS with an equilibrium
spread of 2% and a NP of $1 would be $0.084247:
5
(.02)($1)
PV(CDS Payments
)
 $0.084247
t
t 1 (1.06)
38
CDS Valuation
 The buyer (seller) of this 5-year CDS would
therefore be willing to make (receive) payments
over five years that have a present value of
$0.084247.
39
CDS Valuation
 Because the spread can also be viewed as an
expected loss of principal, the present value of the
payments is also equal to the expected default
protection the buyer (seller) receives (pays).
 The value of the CDS protection, in turn, is equal
to the present value of the expected payout in the
case of default.
40
CDS Valuation
 The present value
of the expected
payout in the case
of default:
where:
pt NP(1  RR )
PV(ExpectedPayout)  
t
(
1

R
)
t 1
M
• pt = probability of default in period t conditional
on no earlier default.
• RR = recovery rate (as a proportion of the face
value) on the bond at the time of default.
• NP = notional principal equal to the par value of
the bond.
41
CDS Valuation
 Note that the probability of default, pt, is defined as
a conditional probability of no prior defaults.
 Thus the conditional probability of default in
Year 4 is based on the probability that the bond
will survive until Year 4.
 In contrast, an unconditional probability is the
likelihood that the bond will default at a given time
as seen from the present.
42
CDS Valuation
 As noted in Chapter 5, conditional default
probabilities are referred to as default intensities.
 Over a period of time, these probabilities will
change, increasing or decreasing depending on the
quality of the credit.
43
CDS Valuation
 Instead of defining a CDS’s expected payout in terms
periodic probability density, pt, the CDS’s expected payout
can alternatively by defined by the average conditional
default probability, p :
PV(ExpectedPayout) 
p NP(1  RR )

(1  R ) t
t 1
M
M
PV(ExpectedPayout)  pNP(1  RR )
t 1
1
(1  R ) t
44
CDS Valuation
 Given the equilibrium spread of .02 in our example
and assuming a recovery rate of 30% if the
underlying bond defaults, the implied probability
density for our illustrative CDS would be .02857.
 This implied probability is obtained by solving for
the p that makes the present value of the expected
payout equal to present value of the payments of
$.084247.
45
CDS Valuation
P V(ExpectedP ayout)  P V(P ayments)
p NP(1  RR ) M Z NP
 (1  R )t   (1  R )t
t 1
t 1
M
 The implied probability
is obtained by solving
for the p that makes
the present value of the
expected payout equal
to present value of the
payments:
Z
(1  RR )
.02
p 
 .02857
(1  .30)
p 
p NP(1  RR )
t
(
1

R
)
t 1
M
P V(ExpectedP ayout)  
P V(ExpectedP ayout) 
(.02857) ($1)(1  .30)

t
(
1
.
06
)
t 1
5
P V(ExpectedP ayout)  $0.084247
46
CDS Valuation
 Note that if there were no recovery (RR = 0), then
the implied probability would be equal to the spread
Z, which as noted can be thought of as the
probability of default of principal.
 The probability density implied by the market is
referred to as the risk-neutral probability because it
is based on an equilibrium spread that is arbitrage
free.
47
Alternative CDS Valuation
48
Alternative CDS
Valuation Approach
 Suppose in the illustrative example, the
estimated default intensity, sometimes referred
to as the real world probability, on the 5-year
BBB bond were .02 and not the implied
probability of .02857.
49
Alternative CDS
Valuation Approach
 In this case, the present value of the CDS expected
payout would be $0.058973 instead of $0.084247:
(.02) ($1)(1  .30)
 $0.058973
t
(1.06)
t 1
5
PV(ExpectedPayout)  
50
Alternative CDS
Valuation Approach
 Given the spread on the CDS is at 2% and the
present value of the payments are $0.084247,
buyers of the CDS would have to pay more on the
CDS than the value they receive on the expected
payoff ($0.058973).
 If the real world probability density of .02 is
accurate, then buyers of the CDS would eventually
push the CDS spread down until it is equal to the
value of the protection.
51
Alternative CDS
Valuation Approach
 For the payment on the CDS to match the expected
protection, the spread would have to equal .014.
 This implied spread is found by solving for the Z
that equates the present value of the payments to the
present value of the expected payout given the real
world probability of .02 and the estimated recovery
rate of RR = .30.
52
Alternative CDS
Valuation Approach
P V(P ayment)s  P V(Expect edP ayout)
M
 The spread, Z, that
equates the present
value of the payments to
the present value of the
expected payout given
the real world
probability is .014.
Z NP
 (1  R )t 
t 1
p NP(1  RR )
 (1  R )t
t 1
M
Z  p (1  RR )
Z  (.02)(1  .30)  .014
M
Z NP
P V(P ayment)s  
t
(
1

R
)
t 1
5
(.014) ($1)
P V(P ayment)s  
t
(
1
.
06
)
t 1
P V(P ayment)s  $0.058973
53
Comparison of Valuation Approach
 We now have two alternative methods for
pricing a CDS.
54
Comparison of Valuation Approach
1.
We can value the CDS swap given the credit spread
in the market and then determine the present value
of the payments;

In terms of our example, we would use the market
spread of 2% and value the swap at $0.084247 with
the implied probability density (or risk-neutral
probability) being .02857.

See Slide 46.
55
Comparison of Valuation Approach
2.
We can value the swap given estimated probabilities
of default and then determine the present value of
the expected payout.

In terms of our example, we would use the estimated
real world probability of .02 and value the CDS at
$0.058973 with the implied credit spread being .014.

See Slide 53.
56
Comparison of Valuation Approach
What valuation method should be
used?
57
Comparison of Valuation Approach
 Some scholars argue for the use of valuing swaps
with risk-neutral probabilities or equivalently
pricing swaps in terms of credit spreads on the
underlying bonds because it reflects an arbitragefree price.
 Thus, in our example they would price the 5-year
CDS equal to the 2% spread with a total value of
$0.084247.
58
Comparison of Valuation Approach
 If one can estimate real-world probabilities on bonds
and CDS that are more accurate than the probabilities
implied by current credit spread, then eventually the
bond and CDS market will price such claims so that
the credit spread reflects the real world probability.
 If this is the case, then in our example one should
price the 5-year CDS at $0.058973 given the estimated
probability density of .02.
 The argument for pricing CDS using real world
probabilities depends on the ability of practitioners to
estimated default probabilities.
59
Estimating Default Rates and Valuing CDS
Based on Estimated Default Intensities
 There are several approaches for estimating
conditional probabilities.
 The simplest and most direct one is to estimate the
probabilities based on historical default rates.
60
Estimating Probability Intensities
from Historical Default Rates

The next slide shows three different probabilities for
corporate bonds with quality ratings of Aaa, Baa, B,
and Caa:
1. Cumulative default rates
2. Unconditional probability rates
3. Conditional probability rates (probability
intensities)
61
Cumulative Default Rates, Probability Intensities,
and CDS Values and Spreads
p t (1  RR )
(1  R ) t
Z  t 1M
1
 (1  R )t
t 1
M
p NP(1  RR )
 t (1  R )t
t 1
M

62
Estimating Probability Intensities
from Historical Default Rates

The probabilities shown in the table are the average historical
cumulative default rates from 1970-2006 as complied by
Moody’s.

The unconditional probability of a bond defaulting during year
t is equal to the difference in the cumulative probability in year
t minus the cumulative probability of default in year t −1.

Example: The probability of a Caa bond default during year 4
is equal to 7.18% (= 46.9%% − 39.72%).
63
Estimating Probability Intensities
from Historical Default Rates

As previously noted, the conditional probability is the probability of
default in a given year conditional on no prior defaults.

This probability is equal to unconditional probability of default in
time t as a proportion of the bond’s probability of survival at the
beginning of the period.

The probability of survival is equal to 100 minus the cumulative
probability.

Example: The probability that a Caa bond will survive until the end
of year three is 39.72% (100 minus its cumulative probability
39.72%), and the probability that the Caa bond will default during
year 4 conditional on no prior defaults is 11.91% (= 7.18%/60.28%).
64
Valuing CDS Based on Estimated
Historical Default Intensities

Using the conditional probabilities generated from the historical
cumulative default rates, the values and spreads for four CDS
with quality ratings of Aaa, Baa, B, and Caa are shown in Slide
62.

Each swap is assumed to have a maturity of five years, annual
payments, NP of $100, and recovery rate of 30%.

The values are obtained by determining the present values of the
expected payoff, with the discount rate assumed to be 6% and
with the possible defaults assumed to occur at the end each year
(implying there are no accrued payments).

The spreads on the CDS are the spreads that equate the present
value of the payments to the present value of the expected
payoff.
65
Valuing CDS Based on Estimated
Historical Default Intensities

Example:
The present value of the expected payoff for the CDS with a
B quality rating is 17.78:
P V(ExpectedP ayoff) 
p t NP(1  RR )

(1  R ) t
t 1
M
 .0524 .06395 .0647125 .0603905 .0608082
P V(ExpectedP ayoff)  ($100)(1  .3) 




2
3
4
(1.06)
(1.06)
(1.06) 5 
 (1.06) (1.06)
P V(ExpectedP ayoff)  17.78
66
Valuing CDS Based on Estimated
Historical Default Intensities
M
Example:

The spread on the Bquality CDS that
equates the present
value of its payments to
the expected payoff of
$17.78 is .0422:
Z NP


t
t 1 (1  R )
p t NP(1  RR )

(1  R ) t
t 1
M
5
$100
 $17.78
t
t 1 (1.06)
$17.78
$17.78
Z 5

 .0422
$100
$421.2364

t
t 1 (1.06)
Z
67
Valuing CDS Based on Estimated
Historical Default Intensities


Example:
As shown in Slide 62, the present values of the
expected payoffs on the Caa quality CDS is
$41.43 and its spread is 0.0983.
As expected, the CDS values and spreads are
greater, the greater the default risk.
68
Estimating Expected Default Rates:
Implied Default Rates

The above default rates are based on historical
frequencies.

Past frequencies are often not the best predictors
of futures probabilities.

If the market is efficient such that the prices of
bonds reflect the market’s expectation of future
economic conditions, then the implied
probabilities based on CDS’s risk spread would
represent an expected future default probability.
69
Estimating Expected Default Rates:
Implied Default Rates

We previously calculated the average implied probability by
solving for the probability that equated the present value of
the expected payoff to the present value of the payments based
on the current bond’s credit spread.

Using this methodology, one could estimate the implied
conditional probabilities for each year for a given quality
rating using the CDS spread on 1-year to m-year swaps.

This would result in a set of implied default probabilities that
could be used to determine the spread on any swap.
70
Estimating Expected Default Rates:
Implied Default Rates

The table shows the spreads (Z) and the implied probability
densities given an assume recovery rate of 30% on 5 B-rated CDS
with maturities ranging from 1 to 5 years:

The implied probability densities are equal to Z(1 −RR).
Maturity
t
1
2
3
4
5
Spread
Implied Probabilty
0.0400
0.0425
0.0450
0.0475
0.0500
0.0571
0.0607
0.0643
0.0679
0.0714
71
Estimating Expected Default Rates:
Implied Default Rates

Given the implied probabilities, the value of a 5-year CDS
based on its expected payoff would be $18.83 and its
spread would be .0447:
P V(Expect edP ayoff) 
p t NP(1  RR )

(1  R ) t
t 1
M
 .0571 .0607 .0643 .0679 .0714 
P V(Expect edP ayoff)  ($100)(1  .3) 




2
3
4
(
1
.
06
)
(
1
.
06
)
(
1
.
06
)
(
1
.
06
)
(1.06) 5 

P V(Expect edP ayoff)  $18.33
M
Z NP


t
(
1

R
)
t 1
p t NP(1  RR )

(1  R ) t
t 1
M
5
$100
 $18.83
t
(
1
.
06
)
t 1
$18.83
$18.83
Z 5

 .0447
$100
$421.2364

t
t 1 (1.06)
Z
72
Summary of the Two
Valuation Approaches
1.

Pricing CDS in Terms Credit Spreads
Many scholars argue for valuing CDS in terms
credit spreads on the underlying bonds because it
result in an arbitrage-free price.

Pricing CDS in terms of bond credits spreads also
implies that the default probability for determining
the expected payout by the seller is a probability
implied by the credit spread of bonds traded in the
market and not a real-world estimated probability.
73
Summary of the Two
Valuation Approaches
2.


Pricing CDS Based on Estimates of the
Conditional Probability
The alternative to pricing CDS in terms of bond
credit spreads is to determine the spread that will
equate the present value of the payments to the
present value of the expected payout based on
estimates of the conditional probability.
Default probabilities can be estimated using
historical cumulative default rates and implied
default rates.
74
Summary of the Two
Valuation Approaches


Note:
There are a number of other more advanced
estimating techniques that practitioners can use to
determine default probabilities.
Of particular note is Gaussian Copula Model.
75
The Value of an
Off-Market CDS Swap
76
The Value of an Off-Market CDS Swap

Similar to a generic par value interest rate swap,
the swap rate on a new CDS is generally set so
that there is not an initial exchange of money.

Over time and as economic conditions change the
credit spreads on new CDS, the value of an
existing CDS will change.
77
The Value of an Off-Market CDS Swap
Example:

Suppose one year after a bond fund manager bought
our illustrative 5-year CDS on BBB bond at the 2%
spread, the economy became weaker and credit
spreads on 4-year BBB bonds and new CDS on such
bonds were 50 bp greater at 2.5%.

Assume for this discussion that CDS spreads are
determine by bond credit spreads in the market.
78
The Value of an Off-Market CDS Swap
Example:

Suppose the bond fund manager sold her 2%
CDS to a swap bank who hedged the CDS by
selling a new 2.5% CDS on the 4-year BBB
bond.

With a buyer’s position on the assumed 2% CDS
and seller’s position on the 2.5% CDS, the swap
bank, in turn, would gain 0.5% of the NP for the
next four years.
79
The Value of an Off-Market CDS Swap
Example:

Given a discount rate of 6%, the present value of this gain
would be $1.73 per $100 NP. The swap banks would
therefore pay the bond manager a maximum of $1.73 for
assuming the swap.
Offsetting Swap Positions
• Buyer of 2% CDS Swap
• Seller of 2.5% CDS Swap
• Pay 2% of NP
• Receive 2.5%
• Receive Default Protection
• Pay Default Protection
Receive 0.5%
per year
SV 
SV 
(Current Spread  Exisitng Spread)(NP)

(1  R ) t
t 1
4
4

t 1
0.005($100)
 $1.73
(1.06) t
80
The Value of an Off-Market CDS Swap

With four years left on the current swap, the increase in
credit spread in the market has increased the value of
the buyer’s position on the CDS swap by $1.73 from
$6.93 to 8.66:
Exisit ng CDS : P V(CDS P ayment)s 
4
Current P V(CDS P ayment)s  
t 1
4
(.02)($100)
 $6.93

t
(1.06)
t 1
(.025)($100)
 $8.66
t
(1.06)
Change in Value  $1.73
81
The Value of an Off-Market CDS Swap

The increase in value on the buyer’s position of
the exiting swap reflects the fact that with poorer
economic conditions, the 2% swap payments
now provide greater default protection (i.e., the
present value of the expect payout is greater).
82
The Value of an Off-Market CDS Swap
 For the initial seller, an increase in credit
spreads causes a decrease in the value on
seller’s positions.
83
The Value of an Off-Market CDS Swap

Example
Suppose that an insurance company was the one
who sold the 5-year CDS on the BBB bond at
the 2% spread to the bond portfolio manager (via
a swap bank) and that one year later the credit
spread on new 4-year CDS on BBB bonds was
again at 2.5%.
84
The Value of an Off-Market CDS Swap


Example:
If the insurance company were to sell its seller’s
position to a swap bank, the swap bank could
hedge the assumed position by taking a buyer’s
position on a new 4-year, 2.5% CDS on the BBB
bond.
With the offsetting positions, the swap bank
would lose 0.5% of the NP for the next four
years.
85
The Value of an Off-Market CDS Swap

Example:
Given a discount rate of 6%, the present value of this loss
would be $1.73 per $100 NP. The swap banks would therefore
charge the insurance company at least $1.73 for assuming the
seller’s position on the swap:
Offsetting Swap Positions
•Seller of 2% CDS Swap
•Buyer of 2.5% CDS Swap
• Receive 2% of NP
• Pay 2.5%
•Pay Default Protection
•Receive Default Protection
Pay 0.5% per year
SV 
(Current Spread  Exisitng Spread)(NP)

(1  R ) t
t 1
SV 
 0.005($100)
  $1.73

t
(1.06)
t 1
4
4
86
The Value of an Off-Market CDS Swap


Example
For the insurance company, the increase in the
credit spread has decreased the value of their
seller’s position on the CDS swap by $1.73.
That is, for the increase in credit risk, the seller
should be receiving $8.66 instead of $6.93.
87
The Value of an Off-Market CDS Swap

Alternatively stated, the poorer economic
conditions reflected in the greater credit spreads
have increased the probability of default on the
BBB bonds and as a result has increased the
present value of the seller’s expected payoff.
88
The Value of an Off-Market CDS Swap

With the credit spread increasing from 2% to 2.5%, the implied
conditional default rate on the bond has increased from .02857 to
.035714, increasing the present value of the seller’s expected
payout from $6.9302 to $8.66:
Exisitng p 
Z
.02

 .02857
(1  RR ) (1  .30)
Current : p 
Z
.025

 .035714
(1  RR ) (1  .30)
P V(ExpectedP ayout) 
p NP(1  RR )

(1  R ) t
t 1
M
Exisitng: P V(ExpectedP ayout) 
Current : P V(ExpectedP ayout) 
(.02857) ($100)(1  .30)
 $6.93

t
(1.05)
t 1
4
4

t 1
(.035714) ($100)(1  .30)
 $8.66
t
(1.05)
89
The Value of an Off-Market CDS Swap


Summary:
To summarize, an increase in the credit spread
will increase the value of the buyer’s position on
an existing CDS and decrease the seller’s
position.
Just the opposite occurs if economic conditions
improve and credit spreads decrease.
90
Other Credit Derivatives
91
Other Credit Derivatives

The market for CDS has grown dramatically over the
last decade. With that growth there has been an
increase in the creation of other credit derivatives.

The most noteworthy of these other credit derivative
are the
1. Binary swap
2. Credit swap basket
3. CDS forward contracts
4. CDS option contracts
5. Contingent swaps
92
Binary CDS

A binary CDS is identical to the generic CDS
except that the payoff in the case of a default is a
specified dollar amount.

Often the fixed payoff is the principal on the
underlying credit. When this is the case, then the
only difference between the generic and binary
swap is that the generic CDS adjust the payoff by
subtracting the recovery value whereas the binary
CDS does not.
93
Credit Swap Basket

In a basket credit default swap, there is a group
of reference entities or credits instead of one and
there is usually a specified payoff whenever one
of the reference entities defaults.

Typically, after the relevant entry defaults, the
swap is terminated.
94
Credit Swap Basket

Basket CDS can vary by the type of agreement
governing the swap:

Add-up basket CDS provides a payout when any
reference credit in the basket defaults

First-to-default CDS provides a payout only when the
first entry defaults

Second-to-default CDS provides a payout when the
second default occurs

Nth-to-default CDS provides a payout when the nth
credit entry defaults.
95
CDS Forward Contract

A CDS forward contract is a contract to take a
buyer’s position or a seller’s position on a
particular CDS at a specified spread at some
future date.

CDS forward contract provide a tool for locking
in the credit spread on future credit position.
96
CDS Option Contract

A CDS option is an option to buy or sell a
particular CDS at specified swap rate at a
specified future time.

Example: A one-year option to buy a 5-year CDS
on GE for 300 bp.

At expiration, the holder of this option would exercise
her right to take the buyer’s position at 300 bp if current
5-year CDSs on GM were greater than 300bp.

In contrast, she would allow the option to expire and
take the current CDS on GE if it is offered at 300 BP or
less.
97
CAT Bond

Somewhat related to credit risk is the catastrophic
(CAT) risk that insurance companies face in
providing protection against hurricanes,
earthquakes, and other natural disasters.

Insurance companies often hedge CAT risk
through reinsurance.

Note: One CAT hedging product that insurance
companies are increasingly using is the issuance of
CAT bonds.
98
CAT Bond

CAT bonds pay the buyer a higher-than-normal
interest rate. In return for the additional interest,
CAT bondholders agrees to provide protection
for losses from a specified event up to a
specified amount or when the losses exceed a
specified amount.
99
CAT Bond


Example:
An insurance company could issue a CAT bond
with a principal of $200 million against a
hurricane cost exceeding $300 million.
The CAT bondholders would then lose some or
all of their principal if the event occurs and the
cost exceeds $300 million.
100